Note that the above equation is the same as in the PAC model with the exception of a change between the existential quantifiers: in PAC, the focus is on the case where \(\exists C,c > 0 : \forall P \in \mathrm {RE}(\mathcal {H})\)  the guarantee holds (which expresses uniformity), while in the universal setting distribution-dependent constants are allowed.
This subtle change in the definition can make the error-rate
landscape vastly different. As an example, consider the case where
for \(\widehat{h}_n\) , we have that \(\mathrm {er}(\widehat{h}_n) \le C(P)e^{-c(P)n}\) , for
every distribution \(P\) , where \(C(P),c(P)\)  are some
distribution-dependent constants. When we take the pointwise
supremum over all of these infinitely many distributions, it can be the
case that
the resulting function drops as \(C^{\prime }/n\) , where \(C^{\prime }\)  is a distribution-independent constant [1]}.
